Interesting. One of the main objections to their having a solely heraldic/ID function seems to be that they were only used for a relatively short time and aren't seen after other protective components were employed.
"
From the late 13th century, plate protection spread from the knees and elbows to encompass the extremities; square plates called ailettes, which protected the shoulder, made a brief appearance between about 1290 and 1325 before giving way to jointed plate defenses that covered the gap between ......plate and upper-arm" (Encyclopedia Britannica)
Also
The Arador Armour Library and
1066 - Medieval Armour Terminology, but as you say there's no agreement as yet. I'm not sure myself though, that being made of leather or wood automatically disqualifies them from having a protective function - not all medieval protection was metal, by a long chalk. Shields for instance in many cases were made of wood and parchment.
This site has some labelled diagrams of assorted suits of armour etc -
Armour Diagrams
Nice vid Apachewarlord

I think the component you're referring to is something
like the 'besagew' but more comprehensive (see armour dictionary in the "beautifuliron.com link) and in your vid at 2:20-2:25 you can see that there's also a plate above that to prevent the lance point glancing upwards into the neck/face. Similar ones in this vid too -
https://www.youtube.com/watch?v=LO3gooVb400
The besagew I think though is intended to complement a shield rather than replace it entirely, at least in the form it's shown here. So, I don't know. Two points come to mind though - armour changed a lot through the period, it was the 'arms race' of its day in a sense so I guess it's possible that the ailette might have fulfilled two functions maybe at different times.
The other thing I was wondering about is whether modern jousting avoids the more traditional shield because it tends to break the user's arm. That's just guesswork though and other pics I've found show that some modern jousts do use traditional shields.
So I suppose that going back to the question of which is correct, the answer is that they all are, at different times and in different places. It's relevant too that knights had to buy and maintain their own equipment, so while say, a King might have a whole selection of different suits and the latest technology, many would have only had the one for both military and tournament use.